Jeffrey M. Girard
University of Kansas
What is moderation in ANOVA and regression models?
What questions can (and can’t) moderation answer?
What tools can we use to test and probe interactions?
ANOVA Examples: 2x2, 3x2
Regression Examples: (2x2), 2xC, CxC
What are some challenges and opportunities?
Moderation adds contextualization/complexity to a model
It allows an IV’s effect to depend on the values of other IVs
It is usually tested using (bilinear) product terms
There won’t be just one effect of the IV for everyone…
…it depends on who each person is, in terms of other IVs
(It is like a gateway drug for mixed effects modeling)
Categorical-by-Categorical - Does the effect of exercise program on weight loss depend on biological sex?
Categorical-by-Continuous - Does the effect of hours of exercise on weight loss depend on exercise program?
Continuous-by-Continuous - Does the effect of hours of exercise on weight loss depend on the effort put in?
Moderation is not mediation and cannot establish causation
Moderation hypotheses should be specific and falsifiable
R is a cross-platform statistical computing environment
RStudio adds a helpful environment for working with R
The {easystats} package adds statistics functions
Training will increase scores (relative to a control condition)
Training will increase scores for women more than for men
| subject | score | condition | gender |
|---|---|---|---|
| 1 | 5.876475 | training | woman |
| 2 | 4.805460 | control | woman |
| 3 | 3.384664 | training | woman |
| 4 | 4.086933 | control | woman |
| 5 | 4.749709 | control | man |
| 6 | 5.905577 | training | woman |
| 7 | 6.005326 | training | man |
| 8 | 3.208133 | control | woman |
| 9 | 4.838511 | training | woman |
| 10 | 5.127727 | training | woman |
| 11 | 5.158661 | control | woman |
| 12 | 4.321588 | control | woman |
| 13 | 5.582489 | training | man |
| 14 | 6.743740 | training | woman |
| 15 | 5.280495 | control | man |
| 16 | 5.543959 | control | woman |
| 17 | 4.709095 | training | man |
| 18 | 4.827145 | control | man |
| 19 | 6.451114 | training | woman |
| 20 | 6.320117 | training | man |
| 21 | 2.589054 | control | man |
| 22 | 4.720086 | control | woman |
| 23 | 5.071068 | control | man |
| 24 | 7.096972 | training | man |
| 25 | 4.274268 | control | woman |
| 26 | 6.460575 | training | man |
| 27 | 6.456440 | training | man |
| 28 | 6.265848 | control | woman |
| 29 | 4.763044 | control | woman |
| 30 | 3.979886 | training | woman |
| 31 | 5.339553 | control | woman |
| 32 | 6.895974 | training | woman |
| 33 | 4.313938 | training | man |
| 34 | 5.095529 | training | man |
| 35 | 4.712894 | control | woman |
| 36 | 5.342586 | control | man |
| 37 | 4.445261 | training | woman |
| 38 | 4.342722 | control | woman |
| 39 | 3.985711 | control | man |
| 40 | 5.953967 | training | woman |
| 41 | 3.846046 | control | man |
| 42 | 5.853868 | training | man |
| 43 | 5.648286 | training | man |
| 44 | 5.876512 | training | woman |
| 45 | 6.506987 | control | woman |
| 46 | 4.572705 | training | woman |
| 47 | 4.412859 | control | woman |
| 48 | 7.958926 | training | woman |
| 49 | 6.173742 | training | woman |
| 50 | 5.834424 | training | man |
| 51 | 6.609042 | control | woman |
| 52 | 4.384601 | training | woman |
| 53 | 5.010130 | control | woman |
| 54 | 3.947642 | control | woman |
| 55 | 5.933923 | training | woman |
| 56 | 5.058090 | control | man |
| 57 | 2.959927 | control | man |
| 58 | 4.873361 | control | man |
| 59 | 3.594148 | control | man |
| 60 | 4.714534 | training | man |
| 61 | 3.264649 | control | man |
| 62 | 4.588575 | training | woman |
| 63 | 5.089966 | control | woman |
| 64 | 4.163905 | control | man |
| 65 | 7.127579 | training | woman |
| 66 | 6.005201 | training | man |
| 67 | 4.437437 | control | man |
| 68 | 5.365165 | control | woman |
| 69 | 4.105399 | training | man |
| 70 | 5.635135 | control | woman |
| 71 | 5.022247 | training | woman |
| 72 | 4.180026 | control | woman |
| 73 | 6.752384 | training | man |
| 74 | 4.181650 | training | woman |
| 75 | 6.044261 | training | man |
| 76 | 4.213103 | control | man |
| 77 | 6.115964 | training | woman |
| 78 | 5.527988 | control | woman |
| 79 | 5.734455 | training | man |
| 80 | 5.387702 | training | man |
| 81 | 4.183703 | training | woman |
| 82 | 5.163233 | training | woman |
| 83 | 5.478410 | control | woman |
| 84 | 3.306088 | control | man |
| 85 | 6.261406 | control | woman |
| 86 | 5.536015 | training | woman |
| 87 | 6.881067 | control | woman |
| 88 | 4.633545 | training | woman |
| 89 | 4.505152 | training | woman |
| 90 | 3.482766 | control | woman |
| 91 | 5.459823 | training | man |
| 92 | 3.713462 | training | man |
| 93 | 6.456246 | training | man |
| 94 | 4.719504 | control | woman |
| 95 | 5.088813 | training | woman |
| 96 | 3.936570 | control | man |
| 97 | 3.045063 | training | man |
| 98 | 5.812886 | training | man |
| 99 | 3.608662 | control | man |
| 100 | 4.625465 | control | man |
| 101 | 3.716192 | training | woman |
| 102 | 4.130171 | training | man |
| 103 | 4.793977 | control | man |
| 104 | 5.767222 | training | man |
| 105 | 5.087236 | control | woman |
| 106 | 6.551865 | control | woman |
| 107 | 6.295529 | training | man |
| 108 | 5.841340 | training | man |
| 109 | 4.390190 | training | man |
| 110 | 3.357638 | control | man |
| 111 | 4.211005 | control | man |
| 112 | 6.286031 | control | woman |
| 113 | 3.568811 | control | woman |
| 114 | 5.112539 | training | man |
| 115 | 4.997397 | training | woman |
| 116 | 4.467550 | control | man |
| 117 | 4.906547 | control | man |
| 118 | 3.718384 | control | woman |
| 119 | 3.944646 | control | woman |
| 120 | 3.917821 | control | man |
| 121 | 5.728532 | control | woman |
| 122 | 5.281620 | training | man |
| 123 | 5.708913 | control | man |
| 124 | 5.249298 | training | woman |
| 125 | 4.454345 | control | woman |
| 126 | 6.730907 | training | woman |
| 127 | 4.962171 | training | woman |
| 128 | 6.095822 | training | woman |
| 129 | 5.951963 | training | man |
| 130 | 3.735796 | control | man |
| 131 | 2.772235 | control | woman |
| 132 | 5.499271 | training | man |
| 133 | 6.358859 | control | woman |
| 134 | 4.172441 | training | man |
| 135 | 5.749763 | training | woman |
| 136 | 3.839864 | control | woman |
| 137 | 6.754586 | training | woman |
| 138 | 5.222653 | training | man |
| 139 | 5.272819 | training | woman |
| 140 | 3.945108 | control | man |
| 141 | 4.287297 | control | man |
| 142 | 4.365866 | control | man |
| 143 | 5.101909 | training | woman |
| 144 | 4.065691 | control | man |
| 145 | 5.559181 | control | woman |
| 146 | 4.757044 | control | woman |
| 147 | 3.809915 | training | man |
| 148 | 3.104902 | control | man |
| 149 | 6.030397 | control | woman |
| 150 | 3.626316 | control | man |
| 151 | 5.397467 | control | woman |
| 152 | 3.667918 | control | man |
| 153 | 6.056584 | training | man |
| 154 | 4.473312 | control | woman |
| 155 | 3.989827 | control | woman |
| 156 | 4.696250 | control | woman |
| 157 | 3.407245 | control | man |
| 158 | 6.073001 | training | man |
| 159 | 5.562025 | control | woman |
| 160 | 4.917394 | training | woman |
| 161 | 6.520379 | training | woman |
| 162 | 4.077757 | control | man |
| 163 | 5.460912 | training | man |
| 164 | 5.231007 | control | woman |
| 165 | 5.748438 | control | man |
| 166 | 6.326105 | control | woman |
| 167 | 5.451902 | training | man |
| 168 | 4.902787 | training | man |
| 169 | 6.107737 | training | man |
| 170 | 5.220152 | control | man |
| 171 | 5.292647 | training | man |
| 172 | 3.713711 | control | woman |
| 173 | 5.905406 | training | man |
| 174 | 6.075597 | training | woman |
| 175 | 3.740196 | control | man |
| 176 | 4.454490 | training | woman |
| 177 | 4.733459 | control | man |
| 178 | 4.269386 | control | woman |
| 179 | 5.404058 | training | woman |
| 180 | 7.179318 | training | man |
| 181 | 4.113804 | control | man |
| 182 | 6.464315 | training | woman |
| 183 | 4.278438 | control | woman |
| 184 | 4.631153 | control | man |
| 185 | 6.812150 | control | woman |
| 186 | 4.390155 | control | woman |
| 187 | 4.555936 | training | woman |
| 188 | 6.047384 | training | man |
| 189 | 5.327824 | control | woman |
| 190 | 5.106458 | training | woman |
| 191 | 4.484803 | training | woman |
| 192 | 4.873373 | training | man |
| 193 | 6.282335 | training | man |
| 194 | 7.693778 | control | woman |
| 195 | 7.364219 | training | man |
| 196 | 5.578385 | control | man |
| 197 | 4.827206 | control | woman |
| 198 | 3.023665 | control | woman |
| 199 | 4.158985 | control | man |
| 200 | 4.455009 | training | man |
We can use the aov() function to fit simple ANOVAs like this
The formula will be DV ~ IV1 * IV2 for a two-way ANOVA
We will then use model_parameters() to get Type 3 results
Finally, we will estimate_means() and estimate_contrasts()
| Parameter | Sum_Squares | df | Mean_Square | F | p |
|---|---|---|---|---|---|
| (Intercept) | 1443.22 | 1 | 1443.22 | 1612.45 | < .001 |
| condition | 4.16 | 1 | 4.16 | 4.65 | 0.032 |
| gender | 13.77 | 1 | 13.77 | 15.38 | < .001 |
| condition:gender | 9.14 | 1 | 9.14 | 10.22 | 0.002 |
| Residuals | 175.43 | 196 | 0.90 |
| condition | gender | Mean | SE | 95% CI |
|---|---|---|---|---|
| control | woman | 4.99 | 0.12 | (4.74, 5.23) |
| training | woman | 5.39 | 0.14 | (5.12, 5.66) |
| control | man | 4.25 | 0.14 | (3.97, 4.53) |
| training | man | 5.51 | 0.13 | (5.25, 5.77) |
Marginal means estimated at condition, gender
| Level1 | Level2 | gender | Difference | 95% CI | SE | t(196) | p |
|---|---|---|---|---|---|---|---|
| control | training | man | -1.26 | (-1.64, -0.88) | 0.19 | -6.48 | < .001 |
| control | training | woman | -0.40 | (-0.77, -0.03) | 0.19 | -2.16 | 0.032 |
Marginal contrasts estimated at condition, p-value adjustment method: Holm (1979)
| Level1 | Level2 | condition | Difference | 95% CI | SE | t(196) | p |
|---|---|---|---|---|---|---|---|
| woman | man | control | 0.74 | ( 0.37, 1.11) | 0.19 | 3.92 | < .001 |
| woman | man | training | -0.12 | (-0.50, 0.26) | 0.19 | -0.64 | 0.526 |
Marginal contrasts estimated at gender, p-value adjustment method: Holm (1979)
Treatment will decrease symptoms (vs. control and placebo)
Treatment will be more effective for women than for men
| patient | symptoms | group | gender |
|---|---|---|---|
| 1 | 3.683067 | treatment | man |
| 2 | 3.831002 | placebo | man |
| 3 | 3.751012 | treatment | man |
| 4 | 3.350506 | treatment | man |
| 5 | 3.999081 | placebo | man |
| 6 | 4.040540 | treatment | man |
| 7 | 3.789107 | placebo | woman |
| 8 | 1.631338 | treatment | woman |
| 9 | 5.865339 | control | man |
| 10 | 3.959936 | placebo | woman |
| 11 | 4.208445 | treatment | man |
| 12 | 5.238969 | control | man |
| 13 | 3.956852 | placebo | woman |
| 14 | 4.101691 | treatment | man |
| 15 | 4.584589 | placebo | woman |
| 16 | 4.654177 | control | woman |
| 17 | 5.096955 | control | woman |
| 18 | 3.757290 | treatment | man |
| 19 | 5.972139 | control | woman |
| 20 | 3.650344 | placebo | man |
| 21 | 3.960412 | placebo | man |
| 22 | 2.371775 | treatment | woman |
| 23 | 4.471384 | control | woman |
| 24 | 4.223687 | treatment | man |
| 25 | 3.720213 | placebo | man |
| 26 | 2.636796 | treatment | woman |
| 27 | 2.561717 | treatment | woman |
| 28 | 3.483105 | placebo | woman |
| 29 | 4.853836 | control | man |
| 30 | 2.742955 | treatment | woman |
| 31 | 2.998217 | treatment | woman |
| 32 | 3.874644 | placebo | woman |
| 33 | 2.410750 | treatment | woman |
| 34 | 4.280384 | control | woman |
| 35 | 3.993854 | placebo | woman |
| 36 | 4.176368 | placebo | man |
| 37 | 5.342086 | control | man |
| 38 | 3.502481 | treatment | man |
| 39 | 2.609248 | treatment | woman |
| 40 | 4.101993 | placebo | woman |
| 41 | 3.875935 | treatment | man |
| 42 | 5.168324 | control | man |
| 43 | 3.295567 | treatment | man |
| 44 | 3.545530 | treatment | man |
| 45 | 5.174403 | control | man |
| 46 | 3.158648 | placebo | woman |
| 47 | 2.672171 | treatment | man |
| 48 | 5.058088 | control | man |
| 49 | 3.227256 | placebo | man |
| 50 | 5.128927 | control | woman |
| 51 | 3.824916 | control | woman |
| 52 | 3.916943 | treatment | man |
| 53 | 3.208675 | treatment | man |
| 54 | 3.885213 | placebo | man |
| 55 | 4.717599 | control | man |
| 56 | 2.829732 | treatment | man |
| 57 | 2.984161 | placebo | woman |
| 58 | 3.834124 | placebo | woman |
| 59 | 3.786460 | treatment | man |
| 60 | 1.984711 | treatment | woman |
| 61 | 3.521415 | treatment | man |
| 62 | 5.283700 | control | man |
| 63 | 3.957941 | placebo | woman |
| 64 | 2.003426 | treatment | woman |
| 65 | 3.302742 | placebo | man |
| 66 | 4.519169 | control | woman |
| 67 | 3.315367 | treatment | man |
| 68 | 3.321480 | placebo | man |
| 69 | 3.092005 | placebo | woman |
| 70 | 4.486210 | control | woman |
| 71 | 3.889311 | control | woman |
| 72 | 3.239938 | placebo | woman |
| 73 | 2.494876 | placebo | woman |
| 74 | 4.865170 | control | man |
| 75 | 3.099088 | placebo | man |
| 76 | 2.703681 | treatment | woman |
| 77 | 3.294807 | treatment | man |
| 78 | 3.167455 | placebo | woman |
| 79 | 4.626240 | control | man |
| 80 | 3.111319 | treatment | woman |
| 81 | 5.244846 | control | woman |
| 82 | 2.789232 | treatment | woman |
| 83 | 4.391974 | placebo | man |
| 84 | 2.671924 | treatment | woman |
| 85 | 4.258345 | control | man |
| 86 | 3.574922 | treatment | man |
| 87 | 4.862326 | control | man |
| 88 | 3.094355 | treatment | woman |
| 89 | 3.424623 | treatment | man |
| 90 | 3.275555 | treatment | woman |
| 91 | 2.438367 | treatment | woman |
| 92 | 3.503636 | placebo | man |
| 93 | 4.049056 | placebo | man |
| 94 | 4.420680 | control | woman |
| 95 | 3.436439 | treatment | man |
| 96 | 3.354488 | placebo | woman |
| 97 | 3.983255 | control | man |
| 98 | 5.182548 | control | man |
| 99 | 2.613542 | placebo | woman |
| 100 | 4.055499 | control | woman |
| 101 | 2.437038 | treatment | woman |
| 102 | 4.679464 | control | man |
| 103 | 4.759928 | placebo | man |
| 104 | 3.029085 | treatment | woman |
| 105 | 5.313118 | control | woman |
| 106 | 5.176724 | control | woman |
| 107 | 4.085248 | control | man |
| 108 | 5.140998 | control | man |
| 109 | 1.968991 | treatment | woman |
| 110 | 5.186876 | control | man |
| 111 | 2.284765 | treatment | woman |
| 112 | 3.658862 | placebo | man |
| 113 | 4.350146 | treatment | man |
| 114 | 4.300288 | treatment | man |
| 115 | 4.686918 | control | man |
| 116 | 3.766755 | control | man |
| 117 | 2.883879 | placebo | woman |
| 118 | 3.178663 | treatment | man |
| 119 | 2.487341 | treatment | woman |
| 120 | 3.659114 | placebo | man |
| 121 | 4.557424 | control | woman |
| 122 | 4.907548 | control | man |
| 123 | 4.655817 | control | woman |
| 124 | 3.494930 | placebo | woman |
| 125 | 2.574772 | placebo | man |
| 126 | 5.417146 | control | man |
| 127 | 3.325603 | placebo | woman |
| 128 | 2.860576 | treatment | woman |
| 129 | 3.338200 | placebo | man |
| 130 | 3.473366 | placebo | woman |
| 131 | 1.553649 | treatment | woman |
| 132 | 3.361876 | treatment | woman |
| 133 | 5.294677 | control | man |
| 134 | 3.866261 | placebo | woman |
| 135 | 3.340717 | treatment | woman |
| 136 | 5.568056 | control | woman |
| 137 | 4.694963 | control | man |
| 138 | 4.268536 | control | man |
| 139 | 2.279876 | treatment | woman |
| 140 | 4.679343 | placebo | man |
| 141 | 3.470273 | placebo | man |
| 142 | 4.831236 | control | woman |
| 143 | 3.635198 | placebo | woman |
| 144 | 3.779816 | placebo | man |
| 145 | 4.048951 | treatment | man |
| 146 | 3.685560 | treatment | man |
| 147 | 5.342175 | control | man |
| 148 | 4.800316 | control | man |
| 149 | 4.426206 | control | woman |
| 150 | 3.059124 | placebo | man |
| 151 | 4.892211 | control | man |
| 152 | 2.261958 | treatment | woman |
| 153 | 1.477009 | treatment | woman |
| 154 | 5.552239 | control | woman |
| 155 | 4.072153 | treatment | man |
| 156 | 3.828199 | placebo | woman |
| 157 | 2.466217 | treatment | woman |
| 158 | 2.902365 | treatment | woman |
| 159 | 4.066778 | treatment | man |
| 160 | 3.591798 | treatment | man |
| 161 | 4.479610 | control | woman |
| 162 | 3.746472 | placebo | man |
| 163 | 3.409870 | treatment | man |
| 164 | 3.564776 | placebo | man |
| 165 | 2.127127 | treatment | woman |
| 166 | 4.304739 | placebo | woman |
| 167 | 2.373575 | treatment | woman |
| 168 | 2.880832 | placebo | woman |
| 169 | 5.201406 | control | man |
| 170 | 3.455701 | placebo | woman |
| 171 | 2.911728 | placebo | woman |
| 172 | 3.925479 | treatment | woman |
| 173 | 3.338932 | placebo | woman |
| 174 | 3.667183 | placebo | woman |
| 175 | 1.314828 | treatment | woman |
| 176 | 2.455877 | treatment | woman |
| 177 | 2.362176 | treatment | woman |
| 178 | 5.052475 | control | man |
| 179 | 5.529821 | control | woman |
| 180 | 3.213364 | placebo | man |
| 181 | 4.841677 | control | man |
| 182 | 3.651969 | placebo | woman |
| 183 | 2.557120 | treatment | woman |
| 184 | 4.788895 | control | woman |
| 185 | 4.023404 | placebo | woman |
| 186 | 4.282011 | control | woman |
| 187 | 2.442558 | placebo | woman |
| 188 | 3.762763 | placebo | woman |
| 189 | 3.966392 | treatment | man |
| 190 | 1.683476 | treatment | woman |
| 191 | 5.951345 | control | man |
| 192 | 2.663725 | treatment | woman |
| 193 | 3.530330 | treatment | man |
| 194 | 4.012764 | treatment | man |
| 195 | 2.412654 | placebo | man |
| 196 | 1.329439 | treatment | woman |
| 197 | 4.522039 | control | woman |
| 198 | 2.634963 | treatment | woman |
| 199 | 5.169679 | control | woman |
| 200 | 3.799478 | treatment | man |
| Parameter | Sum_Squares | df | Mean_Square | F | p |
|---|---|---|---|---|---|
| (Intercept) | 615.35 | 1 | 615.35 | 2364.66 | < .001 |
| group | 86.82 | 2 | 43.41 | 166.81 | < .001 |
| gender | 0.36 | 1 | 0.36 | 1.39 | 0.240 |
| group:gender | 13.09 | 2 | 6.54 | 25.15 | < .001 |
| Residuals | 50.48 | 194 | 0.26 |
| group | gender | Mean | SE | 95% CI |
|---|---|---|---|---|
| control | woman | 4.77 | 0.10 | (4.58, 4.97) |
| placebo | woman | 3.50 | 0.09 | (3.33, 3.67) |
| treatment | woman | 2.48 | 0.08 | (2.33, 2.64) |
| control | man | 4.93 | 0.09 | (4.75, 5.11) |
| placebo | man | 3.63 | 0.10 | (3.44, 3.82) |
| treatment | man | 3.68 | 0.09 | (3.51, 3.84) |
Marginal means estimated at group, gender
| Level1 | Level2 | gender | Difference | 95% CI | SE | t(194) | p |
|---|---|---|---|---|---|---|---|
| control | placebo | man | 1.30 | ( 0.98, 1.62) | 0.13 | 9.81 | < .001 |
| control | placebo | woman | 1.27 | ( 0.96, 1.59) | 0.13 | 9.73 | < .001 |
| control | treatment | man | 1.25 | ( 0.96, 1.55) | 0.12 | 10.20 | < .001 |
| control | treatment | woman | 2.29 | ( 1.99, 2.60) | 0.13 | 18.23 | < .001 |
| placebo | treatment | man | -0.04 | (-0.36, 0.27) | 0.13 | -0.35 | 0.936 |
| placebo | treatment | woman | 1.02 | ( 0.74, 1.30) | 0.12 | 8.76 | < .001 |
Marginal contrasts estimated at group, p-value adjustment method: Holm (1979)
| Level1 | Level2 | group | Difference | 95% CI | SE | t(194) | p |
|---|---|---|---|---|---|---|---|
| woman | man | control | -0.16 | (-0.42, 0.11) | 0.13 | -1.18 | 0.240 |
| woman | man | placebo | -0.13 | (-0.39, 0.13) | 0.13 | -0.98 | 0.327 |
| woman | man | treatment | -1.20 | (-1.42, -0.97) | 0.12 | -10.32 | < .001 |
Marginal contrasts estimated at gender, p-value adjustment method: Holm (1979)
Recreate the results from ANOVA
Incorporate continuous predictors/IVs
Incorporate any kind/number of “covariates”
Incorporate nonlinearity (e.g., polynomials)
Extend to GLM for non-normal outcomes/DVs
Extend to MLM for clustered/nested data
Extend to SEM for latent variables
ANOVA code (for reference)
| Parameter | Sum_Squares | df | Mean_Square | F | p |
|---|---|---|---|---|---|
| (Intercept) | 1443.22 | 1 | 1443.22 | 1612.45 | < .001 |
| condition | 4.16 | 1 | 4.16 | 4.65 | 0.032 |
| gender | 13.77 | 1 | 13.77 | 15.38 | < .001 |
| condition:gender | 9.14 | 1 | 9.14 | 10.22 | 0.002 |
| Residuals | 175.43 | 196 | 0.90 |
| Parameter | Coefficient | SE | 95% CI | t(196) | p |
|---|---|---|---|---|---|
| (Intercept) | 4.99 | 0.12 | (4.74, 5.23) | 40.16 | < .001 |
| condition (training) | 0.40 | 0.19 | (0.03, 0.77) | 2.16 | 0.032 |
| gender (man) | -0.74 | 0.19 | (-1.11, -0.37) | -3.92 | < .001 |
| condition (training) × gender (man) | 0.86 | 0.27 | (0.33, 1.39) | 3.20 | 0.002 |
Note the same \(p\)-values and that each \(F\) is equal to the corresponding \(t\) squared.
By giving the lm() model to the same functions, we can…
Estimate and plot (the same) marginal means
Exercising for a longer duration will burn more calories
Long swims will be more effective than long runs
| participant | calories | duration | exercise |
|---|---|---|---|
| 1 | 486.9765 | 2.0247982 | swim |
| 2 | 472.8959 | 1.3821725 | run |
| 3 | 381.9000 | 1.3681240 | swim |
| 4 | 384.2023 | 1.4285675 | run |
| 5 | 459.0924 | 2.3272059 | run |
| 6 | 539.3930 | 1.6615903 | swim |
| 7 | 401.4003 | 1.4630448 | swim |
| 8 | 397.0844 | 1.5121709 | run |
| 9 | 246.7168 | 1.4368627 | swim |
| 10 | 272.5706 | 1.0153436 | swim |
| 11 | 364.9235 | 1.7090055 | run |
| 12 | 298.7456 | 1.1187464 | run |
| 13 | 517.4730 | 1.7360870 | swim |
| 14 | 238.2022 | 1.3128362 | swim |
| 15 | 273.5526 | 1.0331437 | run |
| 16 | 696.8010 | 2.4898915 | run |
| 17 | 396.4383 | 1.2521552 | swim |
| 18 | 711.3158 | 2.8504048 | run |
| 19 | 421.4667 | 1.2994872 | swim |
| 20 | 520.9031 | 1.7577685 | swim |
| 21 | 399.1551 | 1.3919107 | run |
| 22 | 490.5664 | 2.0523853 | run |
| 23 | 461.3814 | 1.8787314 | run |
| 24 | 418.8214 | 1.2921095 | swim |
| 25 | 386.2042 | 1.4752701 | run |
| 26 | 553.7873 | 2.1345285 | swim |
| 27 | 459.1941 | 1.7606419 | swim |
| 28 | 359.6371 | 1.8502758 | run |
| 29 | 591.2767 | 2.5190743 | run |
| 30 | 648.0497 | 2.5144626 | swim |
| 31 | 311.0183 | 1.0141624 | run |
| 32 | 590.6766 | 1.3583350 | swim |
| 33 | 554.7981 | 2.0242906 | swim |
| 34 | 753.9970 | 2.6053155 | swim |
| 35 | 410.9739 | 1.7576647 | run |
| 36 | 405.6490 | 1.6033452 | run |
| 37 | 504.7794 | 2.0862072 | swim |
| 38 | 703.8329 | 2.5526066 | run |
| 39 | 520.8601 | 2.3704378 | run |
| 40 | 496.4263 | 1.8792767 | swim |
| 41 | 592.9068 | 2.3074787 | run |
| 42 | 688.9164 | 2.6912732 | swim |
| 43 | 459.9332 | 1.7083987 | swim |
| 44 | 604.1794 | 1.9010350 | swim |
| 45 | 442.8850 | 1.6983853 | run |
| 46 | 407.9747 | 1.4556758 | swim |
| 47 | 480.6806 | 2.0919576 | run |
| 48 | 745.8023 | 2.6535681 | swim |
| 49 | 489.1903 | 1.9158584 | swim |
| 50 | 658.7930 | 2.1719204 | swim |
| 51 | 525.0054 | 2.1882373 | run |
| 52 | 553.2563 | 1.9027302 | swim |
| 53 | 225.7835 | 0.9423320 | run |
| 54 | 373.1410 | 1.5434663 | run |
| 55 | 649.7859 | 2.1748546 | swim |
| 56 | 465.0962 | 2.2027886 | run |
| 57 | 524.8834 | 2.2526631 | run |
| 58 | 298.0463 | 1.1040664 | run |
| 59 | 503.7600 | 1.6692555 | run |
| 60 | 535.0446 | 1.8138636 | swim |
| 61 | 453.8953 | 2.0793305 | run |
| 62 | 448.7485 | 1.6607938 | swim |
| 63 | 522.9352 | 2.0412445 | run |
| 64 | 611.5877 | 2.6218701 | run |
| 65 | 693.7688 | 2.4402478 | swim |
| 66 | 494.5699 | 2.2719794 | swim |
| 67 | 436.5890 | 1.6045474 | run |
| 68 | 506.5195 | 2.2135723 | run |
| 69 | 657.3150 | 2.4755572 | swim |
| 70 | 566.0532 | 2.4100585 | run |
| 71 | 292.0450 | 1.0945270 | swim |
| 72 | 474.1026 | 1.8600432 | run |
| 73 | 617.6613 | 2.3355341 | swim |
| 74 | 816.7769 | 2.7984863 | swim |
| 75 | 462.7039 | 1.6371341 | swim |
| 76 | 551.6663 | 2.4802876 | run |
| 77 | 713.1180 | 2.4782198 | swim |
| 78 | 609.7088 | 2.6329238 | run |
| 79 | 476.8087 | 1.8815219 | swim |
| 80 | 370.2146 | 1.2399430 | swim |
| 81 | 668.3936 | 2.1697767 | swim |
| 82 | 686.7034 | 2.6979870 | swim |
| 83 | 429.6422 | 1.4069689 | run |
| 84 | 442.3489 | 1.7977642 | run |
| 85 | 425.3900 | 1.8564469 | run |
| 86 | 672.3602 | 2.4712929 | swim |
| 87 | 372.3098 | 1.4726307 | run |
| 88 | 456.4376 | 1.6713612 | swim |
| 89 | 563.7933 | 1.7928556 | swim |
| 90 | 496.6736 | 2.2269833 | run |
| 91 | 509.8858 | 1.7230230 | swim |
| 92 | 601.9327 | 2.1769343 | swim |
| 93 | 485.9239 | 2.0741429 | swim |
| 94 | 510.3737 | 2.1882560 | run |
| 95 | 750.1839 | 2.7534936 | swim |
| 96 | 367.3109 | 1.5363525 | run |
| 97 | 497.9109 | 1.7064294 | swim |
| 98 | 840.7540 | 3.2294629 | swim |
| 99 | 616.4693 | 2.3368709 | run |
| 100 | 466.5235 | 2.1672122 | run |
| 101 | 813.8330 | 2.8045210 | swim |
| 102 | 341.9197 | 1.4423004 | swim |
| 103 | 538.2966 | 2.0050650 | run |
| 104 | 533.4335 | 1.4738210 | swim |
| 105 | 566.0748 | 2.2169616 | run |
| 106 | 590.7031 | 2.3290448 | run |
| 107 | 338.1217 | 1.2799633 | swim |
| 108 | 610.8300 | 2.2366806 | swim |
| 109 | 438.1448 | 1.5970739 | swim |
| 110 | 387.2061 | 1.6072670 | run |
| 111 | 310.1953 | 1.4323247 | run |
| 112 | 376.5662 | 1.5442874 | run |
| 113 | 479.5830 | 2.0449831 | run |
| 114 | 480.1373 | 1.8819526 | swim |
| 115 | 724.9867 | 2.8137896 | swim |
| 116 | 604.7393 | 2.2526003 | run |
| 117 | 542.7069 | 2.0187187 | run |
| 118 | 504.2774 | 2.1825826 | run |
| 119 | 348.8282 | 1.3026996 | run |
| 120 | 519.7228 | 2.3175673 | run |
| 121 | 462.2700 | 1.7611237 | run |
| 122 | 408.7308 | 1.5900129 | swim |
| 123 | 582.0695 | 2.6261921 | run |
| 124 | 411.2592 | 1.3408250 | swim |
| 125 | 567.4711 | 2.2721306 | run |
| 126 | 495.5337 | 1.9065515 | swim |
| 127 | 685.5443 | 2.3079818 | swim |
| 128 | 581.1644 | 2.2639938 | swim |
| 129 | 512.6728 | 2.1172276 | swim |
| 130 | 533.8828 | 1.9438510 | run |
| 131 | 334.7888 | 1.3418515 | run |
| 132 | 539.4534 | 1.8316162 | swim |
| 133 | 505.0998 | 2.2392051 | run |
| 134 | 388.8688 | 1.4530442 | swim |
| 135 | 649.0873 | 2.6307028 | swim |
| 136 | 513.1374 | 2.0180076 | run |
| 137 | 834.2074 | 2.9405336 | swim |
| 138 | 437.2265 | 1.5667725 | swim |
| 139 | 503.4449 | 1.5025761 | swim |
| 140 | 345.7169 | 1.2413832 | run |
| 141 | 539.9918 | 1.9799114 | run |
| 142 | 352.5483 | 1.1067311 | run |
| 143 | 724.8370 | 2.4781231 | swim |
| 144 | 521.8461 | 1.8597521 | run |
| 145 | 412.4266 | 1.7944063 | run |
| 146 | 433.7123 | 1.7682848 | run |
| 147 | 255.0309 | 0.7725315 | swim |
| 148 | 503.2697 | 2.1564432 | run |
| 149 | 369.8479 | 1.6043309 | run |
| 150 | 575.9216 | 2.1127325 | run |
| 151 | 330.4060 | 1.1080961 | run |
| 152 | 345.6405 | 1.3150856 | run |
| 153 | 592.7174 | 2.1969883 | swim |
| 154 | 489.6851 | 2.1336110 | run |
| 155 | 451.7196 | 2.0436179 | run |
| 156 | 622.8804 | 2.7759325 | run |
| 157 | 529.4093 | 2.3977643 | run |
| 158 | 578.7621 | 2.1706701 | swim |
| 159 | 377.5654 | 1.4450951 | run |
| 160 | 403.5881 | 1.4788189 | swim |
| 161 | 566.3034 | 1.9055026 | swim |
| 162 | 571.0665 | 2.6430154 | run |
| 163 | 426.0878 | 1.2844053 | swim |
| 164 | 400.3703 | 1.8062697 | run |
| 165 | 473.1179 | 1.7486985 | run |
| 166 | 568.8180 | 2.0337752 | run |
| 167 | 610.9184 | 2.2532732 | swim |
| 168 | 391.3575 | 1.3591922 | swim |
| 169 | 456.8028 | 1.4723230 | swim |
| 170 | 472.4160 | 1.7589106 | run |
| 171 | 677.9135 | 2.3642659 | swim |
| 172 | 456.1863 | 1.8908098 | run |
| 173 | 720.5866 | 2.6544566 | swim |
| 174 | 483.1441 | 1.8746492 | swim |
| 175 | 499.1261 | 1.7271725 | run |
| 176 | 669.3015 | 2.6154537 | swim |
| 177 | 425.2697 | 1.7310856 | run |
| 178 | 587.5653 | 2.2979109 | run |
| 179 | 527.2292 | 2.2259815 | swim |
| 180 | 502.5024 | 1.6678979 | swim |
| 181 | 266.1488 | 0.8861173 | run |
| 182 | 584.7520 | 1.9996355 | swim |
| 183 | 570.9862 | 2.6794293 | run |
| 184 | 363.5350 | 1.3362204 | run |
| 185 | 528.4117 | 2.1248815 | run |
| 186 | 270.5534 | 1.4199320 | run |
| 187 | 607.8367 | 2.6272930 | swim |
| 188 | 502.5881 | 1.8613267 | swim |
| 189 | 490.5813 | 1.8864095 | run |
| 190 | 436.3197 | 1.7725541 | swim |
| 191 | 575.4885 | 1.9436487 | swim |
| 192 | 534.6002 | 1.9829328 | swim |
| 193 | 499.4114 | 1.8009545 | swim |
| 194 | 469.5905 | 1.8328453 | run |
| 195 | 621.7169 | 2.2795903 | swim |
| 196 | 487.5178 | 1.8785222 | run |
| 197 | 310.7343 | 1.1549574 | run |
| 198 | 340.1598 | 1.3524512 | run |
| 199 | 604.2201 | 2.5151987 | run |
| 200 | 485.3175 | 1.6131578 | swim |
| 201 | 517.7148 | 2.1987332 | run |
| 202 | 345.6452 | 1.6339589 | run |
| 203 | 432.5128 | 2.2782919 | run |
| 204 | 471.3572 | 1.7366560 | run |
| 205 | 469.3288 | 1.4949134 | swim |
| 206 | 500.2970 | 1.8481248 | run |
| 207 | 473.2479 | 1.5036224 | swim |
| 208 | 574.3772 | 2.2865004 | run |
| 209 | 549.5592 | 2.2810127 | run |
| 210 | 457.4513 | 1.7086970 | run |
| 211 | 619.9736 | 2.5101894 | run |
| 212 | 402.9696 | 1.8388787 | run |
| 213 | 523.0114 | 1.9804562 | swim |
| 214 | 534.0091 | 2.1155032 | run |
| 215 | 806.5378 | 2.6742191 | swim |
| 216 | 611.5919 | 2.6630524 | run |
| 217 | 634.6245 | 1.9759511 | swim |
| 218 | 454.8344 | 1.7013932 | swim |
| 219 | 481.7028 | 2.3038684 | run |
| 220 | 629.6170 | 2.4100758 | swim |
| 221 | 558.3575 | 1.8963233 | swim |
| 222 | 327.4352 | 1.3568557 | run |
| 223 | 577.7574 | 2.2027033 | swim |
| 224 | 652.6858 | 2.2877986 | swim |
| 225 | 399.3339 | 1.6700979 | run |
| 226 | 498.4742 | 1.4772452 | swim |
| 227 | 573.9528 | 2.1667294 | swim |
| 228 | 391.2898 | 1.6346929 | run |
| 229 | 447.3620 | 1.9520290 | run |
| 230 | 641.0104 | 2.8396592 | run |
| 231 | 508.5105 | 1.8569021 | run |
| 232 | 539.4666 | 2.4821575 | run |
| 233 | 430.4994 | 1.6392192 | swim |
| 234 | 595.6283 | 2.1155764 | swim |
| 235 | 580.6120 | 2.9060750 | run |
| 236 | 478.5022 | 1.6950776 | swim |
| 237 | 381.8860 | 1.5279678 | run |
| 238 | 569.9180 | 2.2736918 | run |
| 239 | 567.5700 | 2.1639122 | swim |
| 240 | 344.5753 | 1.8032291 | run |
| 241 | 395.8164 | 1.4924014 | swim |
| 242 | 425.6812 | 1.6866865 | swim |
| 243 | 784.5746 | 2.3911675 | swim |
| 244 | 620.0504 | 3.3468893 | run |
| 245 | 692.6378 | 2.9321096 | run |
| 246 | 617.8225 | 2.5891923 | run |
| 247 | 518.3671 | 1.9136030 | run |
| 248 | 250.5662 | 1.0118327 | run |
| 249 | 469.9183 | 1.8794926 | run |
| 250 | 382.4043 | 1.4775046 | swim |
| 251 | 532.7137 | 2.4126705 | run |
| 252 | 216.6400 | 0.9246425 | run |
| 253 | 468.9990 | 2.1378735 | run |
| 254 | 530.0574 | 2.5336028 | run |
| 255 | 602.5167 | 2.8599362 | run |
| 256 | 581.5413 | 2.1499824 | swim |
| 257 | 617.4461 | 2.5497054 | swim |
| 258 | 646.6101 | 2.2573090 | swim |
| 259 | 626.1248 | 2.2447256 | swim |
| 260 | 455.8306 | 1.4956228 | swim |
| 261 | 665.0507 | 2.2969297 | swim |
| 262 | 404.4268 | 1.8882466 | run |
| 263 | 352.5399 | 1.3676008 | run |
| 264 | 552.6479 | 1.9058879 | swim |
| 265 | 566.5589 | 2.1059959 | run |
| 266 | 404.2725 | 1.5409786 | swim |
| 267 | 420.8090 | 1.5301990 | swim |
| 268 | 412.8585 | 1.8357175 | run |
| 269 | 750.7532 | 2.6710935 | swim |
| 270 | 521.4205 | 1.9897437 | run |
| 271 | 550.4266 | 1.8602216 | run |
| 272 | 453.5186 | 1.6679728 | run |
| 273 | 860.0864 | 3.0289471 | swim |
| 274 | 469.8367 | 1.4904957 | run |
| 275 | 657.8459 | 2.5975101 | swim |
| 276 | 452.2333 | 2.1491049 | swim |
| 277 | 480.4230 | 2.0775148 | run |
| 278 | 557.5080 | 2.1295221 | run |
| 279 | 632.9417 | 2.2720281 | swim |
| 280 | 468.6496 | 1.4711412 | swim |
| 281 | 484.7340 | 2.4699356 | run |
| 282 | 443.4077 | 1.8582948 | run |
| 283 | 482.2144 | 1.5864967 | run |
| 284 | 741.5248 | 2.9812446 | swim |
| 285 | 348.4961 | 1.5521005 | run |
| 286 | 747.9164 | 3.2244027 | run |
| 287 | 395.2979 | 1.7333078 | run |
| 288 | 532.4208 | 2.0444902 | run |
| 289 | 413.3065 | 1.2879297 | run |
| 290 | 483.3166 | 1.8916819 | run |
| 291 | 445.5594 | 1.8529907 | swim |
| 292 | 507.4554 | 1.7979370 | swim |
| 293 | 409.4575 | 1.7568796 | swim |
| 294 | 301.9665 | 1.2992459 | run |
| 295 | 523.3978 | 2.1804282 | run |
| 296 | 715.6622 | 2.9466068 | swim |
| 297 | 448.0202 | 1.9812995 | swim |
| 298 | 589.6015 | 1.5747256 | swim |
| 299 | 544.2143 | 2.2365115 | swim |
| 300 | 536.1614 | 2.0824728 | swim |
| Parameter | Coefficient | SE | 95% CI | t(296) | p |
|---|---|---|---|---|---|
| (Intercept) | 101.44 | 16.54 | (68.89, 133.99) | 6.13 | < .001 |
| duration | 191.42 | 8.28 | (175.12, 207.72) | 23.11 | < .001 |
| exercise (swim) | -38.85 | 24.49 | (-87.05, 9.35) | -1.59 | 0.114 |
| duration × exercise (swim) | 52.51 | 12.15 | (28.60, 76.42) | 4.32 | < .001 |
The slope of duration is different for running vs. swimming
We can use estimate_slopes() to get these “simple slopes”
| exercise | Coefficient | SE | 95% CI | t(296) | p |
|---|---|---|---|---|---|
| run | 191.42 | 8.28 | (175.12, 207.72) | 23.11 | < .001 |
| swim | 243.93 | 8.89 | (226.44, 261.42) | 27.45 | < .001 |
Marginal effects estimated for duration
| Level1 | Level2 | duration | Difference | 95% CI | SE | t(296) | p |
|---|---|---|---|---|---|---|---|
| run | swim | 1.00 | -13.66 | ( -39.28, 11.96) | 13.02 | -1.05 | 0.295 |
| run | swim | 2.00 | -66.16 | ( -77.68, -54.65) | 5.85 | -11.31 | < .001 |
| run | swim | 3.00 | -118.67 | (-146.10, -91.24) | 13.94 | -8.51 | < .001 |
Marginal contrasts estimated at exercise, p-value adjustment method: Holm (1979)
Attachment anxiety and avoidance will both predict reduced marital satisfaction
The lowest satisfaction will be from those high on both anxiety and avoidance
| participant | satisfaction | aa_anxiety | aa_avoidance |
|---|---|---|---|
| 1 | 0.4318792 | 0.9001420 | -1.2979397 |
| 2 | 0.7464191 | -1.1733458 | -1.6821959 |
| 3 | 0.3861420 | -0.8974854 | 0.0513762 |
| 4 | 0.2232373 | -1.4445014 | -0.2682647 |
| 5 | -0.7684968 | -0.3310136 | -0.0227639 |
| 6 | 1.0004048 | -2.9006290 | 0.3812566 |
| 7 | -0.3436840 | -1.0592557 | 0.3629539 |
| 8 | -0.4922589 | 0.2779547 | 0.4375966 |
| 9 | -0.1300136 | 0.7494859 | -0.7302396 |
| 10 | -0.3741905 | 0.2415825 | -0.8557735 |
| 11 | 0.1709354 | 1.0061857 | 0.2164843 |
| 12 | -1.1790747 | -0.1851460 | 1.1400816 |
| 13 | 1.6410552 | -0.9818267 | -1.6680926 |
| 14 | -0.9697921 | 0.0929079 | -0.3050043 |
| 15 | 0.8311237 | -0.0527844 | -0.4654157 |
| 16 | 1.4663046 | -0.0803279 | 0.0517803 |
| 17 | 0.3227590 | -0.6541037 | 0.2455527 |
| 18 | 0.5613374 | -0.9506835 | -1.5182153 |
| 19 | 0.2867013 | 1.0195618 | -0.5799413 |
| 20 | 0.2252759 | 0.8590464 | -0.1091493 |
| 21 | 0.4219218 | 0.3644608 | 0.8277526 |
| 22 | -0.4174537 | 0.3836510 | -0.0410328 |
| 23 | -0.8307720 | 1.1134057 | 1.6417072 |
| 24 | -1.4653868 | 1.2115098 | 0.2353753 |
| 25 | 1.6109624 | -0.3483255 | -0.6145876 |
| 26 | -0.1949855 | -0.8595534 | 0.8395645 |
| 27 | -0.4571056 | 0.6500272 | -0.2379615 |
| 28 | 0.3654884 | 0.3280591 | 0.6915284 |
| 29 | -1.2968775 | -0.5179466 | 0.2454411 |
| 30 | 0.8860042 | -0.2389821 | -0.6837880 |
| 31 | 0.5425709 | 0.1177789 | -1.9980682 |
| 32 | 0.1560483 | 0.8315185 | 0.3260024 |
| 33 | -0.4192886 | -1.5589189 | 0.6991556 |
| 34 | 0.5947842 | -0.2205191 | -1.2905544 |
| 35 | 0.6887242 | -0.8171944 | -0.0523127 |
| 36 | -2.3396787 | 1.0766774 | -0.6557311 |
| 37 | -2.9128123 | 1.0796640 | 1.5789708 |
| 38 | -0.3232953 | 0.1421281 | -0.1849386 |
| 39 | 0.4294992 | 0.1569795 | -0.1330408 |
| 40 | -1.0115718 | -0.1687203 | -0.4641793 |
| 41 | 0.9539652 | -0.2690377 | -0.1847743 |
| 42 | -0.7423944 | 0.8077684 | 0.2863153 |
| 43 | 0.6097210 | -1.1247172 | -0.7653544 |
| 44 | 1.1044353 | -1.4307880 | -0.8196022 |
| 45 | -0.0529032 | 0.0603567 | 0.5585867 |
| 46 | 0.9429142 | -0.7929825 | -0.4991149 |
| 47 | 0.0985013 | 0.3402759 | -1.4183955 |
| 48 | 0.0725268 | -0.2594687 | -1.2749390 |
| 49 | 0.7930816 | -1.3048486 | 0.4893368 |
| 50 | 0.4904928 | 0.3681734 | -0.5604107 |
| 51 | -1.5627198 | 1.6931900 | 1.0131452 |
| 52 | -1.8050931 | 0.9958370 | -0.2896957 |
| 53 | -2.5484336 | 0.1867521 | 0.6029505 |
| 54 | 0.0655834 | 1.2383374 | -0.0099835 |
| 55 | 0.8203403 | 0.3093733 | -0.8009166 |
| 56 | 0.5053028 | 0.6357179 | -0.0267354 |
| 57 | 1.0059764 | 0.0231833 | -0.8906945 |
| 58 | -0.4588432 | 1.1778636 | 0.9848438 |
| 59 | 0.1994137 | -0.4535466 | 0.3710756 |
| 60 | 0.4312962 | 0.4159275 | -0.3659960 |
| 61 | 1.1502339 | -1.3384424 | 0.4676402 |
| 62 | -0.3208243 | -1.2919747 | -0.7570802 |
| 63 | -0.2599497 | -0.3090742 | -0.1314944 |
| 64 | 0.2481826 | 0.1565121 | 0.2906056 |
| 65 | 2.2388475 | -0.8339168 | 0.9577650 |
| 66 | -0.5760315 | -0.0245493 | 1.2365699 |
| 67 | 2.4549184 | -1.1373516 | -0.4462885 |
| 68 | -1.0100661 | 1.0720542 | -0.1367433 |
| 69 | -3.3026389 | 2.3144986 | 1.4375254 |
| 70 | -0.8252522 | 0.4229731 | 0.9341494 |
| 71 | 0.7765073 | -0.1369390 | -0.2234494 |
| 72 | -1.0488954 | 1.3283965 | -0.6792860 |
| 73 | -0.7864643 | 0.4365357 | 0.5555319 |
| 74 | 0.5199958 | 0.0664285 | 0.5760184 |
| 75 | -1.0993047 | 1.3046598 | -0.1085759 |
| 76 | 1.7649755 | -0.2092595 | -1.0254949 |
| 77 | -1.0799968 | 1.0182530 | 0.7042143 |
| 78 | -1.1299071 | 1.3660350 | -0.1513658 |
| 79 | -1.6126031 | 1.4766958 | 0.4766445 |
| 80 | -1.5209325 | 0.8873065 | 1.9006730 |
| 81 | 1.5738863 | -1.0159089 | -0.6215937 |
| 82 | -2.6556382 | 1.8708691 | 1.6032157 |
| 83 | -1.1665211 | 1.0761984 | -0.2502308 |
| 84 | 0.9498327 | -1.0744076 | -0.1646951 |
| 85 | -0.7852938 | -2.1955760 | 0.8828853 |
| 86 | -0.1718298 | 0.5345446 | -0.3473039 |
| 87 | -0.7909224 | 1.3437334 | -0.3570569 |
| 88 | -0.6439796 | 1.3850035 | 1.0377893 |
| 89 | -2.5453460 | 2.7469268 | 1.3386057 |
| 90 | -1.9914999 | -0.0459447 | -0.3620045 |
| 91 | -0.8787975 | 0.7430853 | -0.6450919 |
| 92 | -1.0239805 | 0.2604247 | -0.4642506 |
| 93 | 2.3690470 | 0.4282819 | 0.9011336 |
| 94 | -3.2273217 | -0.3682588 | 2.3744850 |
| 95 | -2.7747857 | 2.8874233 | 2.8117409 |
| 96 | 0.2915870 | -0.6070726 | 0.8843984 |
| 97 | 1.9462405 | -1.8791987 | -0.8361123 |
| 98 | -0.6213365 | 0.7183566 | -1.5432852 |
| 99 | -0.0092614 | 0.2514256 | -0.1108923 |
| 100 | -0.9133249 | 0.4670230 | -0.7747928 |
| 101 | -0.8827214 | -0.2178884 | 0.7018330 |
| 102 | -0.1254565 | -1.2176477 | -2.4209004 |
| 103 | -0.5933691 | -0.7540785 | -0.0049251 |
| 104 | -1.7725510 | 0.1912302 | 1.0769280 |
| 105 | -2.5214475 | 1.0472828 | 1.9973757 |
| 106 | -0.8901792 | 1.0898284 | 0.6997152 |
| 107 | -2.0053899 | 0.6954744 | 1.2932863 |
| 108 | 0.9362436 | -0.5243688 | 0.3010202 |
| 109 | 0.0788841 | 0.2640159 | 0.5694408 |
| 110 | 0.0090500 | 1.2412004 | -0.4548316 |
| 111 | 0.3197110 | 0.6432972 | 0.8063503 |
| 112 | -0.6545846 | -0.9967626 | -0.5565229 |
| 113 | 0.9702854 | -2.0022985 | -1.8917533 |
| 114 | 0.8073802 | -0.4758391 | -0.3311469 |
| 115 | 1.0146450 | 0.0441800 | 0.2314547 |
| 116 | 0.1046358 | -1.3811354 | -1.3411793 |
| 117 | -0.3001633 | 0.2475197 | -0.7585245 |
| 118 | -0.9326808 | 0.2241897 | -0.2019526 |
| 119 | -0.1125392 | 0.8614789 | 1.5792732 |
| 120 | 1.2038981 | -0.9267623 | -0.3428551 |
| 121 | 2.0238313 | -0.3204430 | -0.3581328 |
| 122 | 0.3891326 | 0.6730248 | -0.3462100 |
| 123 | 1.8491610 | -1.2656298 | 1.4542733 |
| 124 | 2.1707985 | -0.5185210 | -1.1154073 |
| 125 | -1.7876233 | 1.1224941 | 1.5397199 |
| 126 | -2.5309899 | -0.3159775 | 0.1779485 |
| 127 | 0.4381641 | -1.7582992 | -0.4881459 |
| 128 | -0.0901157 | 1.2372534 | 0.7164108 |
| 129 | 1.2519594 | -1.6639524 | -0.0933916 |
| 130 | 0.0103133 | 1.8557345 | -0.2727467 |
| 131 | -3.0334229 | 1.0579974 | 1.2798458 |
| 132 | -0.3140016 | 0.0372688 | -0.2293450 |
| 133 | 1.7196178 | 0.0046050 | -0.7442523 |
| 134 | 0.1709165 | -1.9438343 | 1.1150912 |
| 135 | -0.0547972 | -1.5696234 | -1.3903439 |
| 136 | -0.1978789 | -0.2732972 | 2.1830224 |
| 137 | -0.9573274 | 0.4688538 | -0.3008839 |
| 138 | -0.0407302 | 1.0325388 | 0.4833596 |
| 139 | 2.3119509 | -1.4183961 | -1.8231016 |
| 140 | 0.4058282 | -0.0878775 | -0.2138833 |
| 141 | -2.3535010 | 1.5786307 | 0.3311409 |
| 142 | 0.0420361 | 0.2446249 | -0.2642851 |
| 143 | -2.4964336 | 1.4970935 | 0.1231685 |
| 144 | -1.0481056 | 0.9159064 | -0.9384831 |
| 145 | -0.3707612 | 0.3208412 | 0.4715169 |
| 146 | -2.0273697 | 0.5627243 | 1.9784298 |
| 147 | -2.2883301 | 0.5392953 | 0.1839594 |
| 148 | 3.5094214 | -1.0734049 | -1.1648886 |
| 149 | -0.2352608 | -1.8831606 | -0.2402030 |
| 150 | -1.3108549 | 0.9113902 | 0.5088021 |
| 151 | -0.2716284 | 0.0495964 | -1.2340717 |
| 152 | -0.2418530 | -1.2356549 | 1.5313468 |
| 153 | 1.5319982 | -1.2637519 | -0.8697801 |
| 154 | -0.1597984 | -1.1428650 | -0.2467974 |
| 155 | 0.8184913 | 0.6544118 | -1.5212871 |
| 156 | 2.8404056 | -0.6768194 | 1.0122899 |
| 157 | 1.2801483 | -1.0739105 | -0.8426715 |
| 158 | 0.1640472 | -0.9756581 | -0.2559447 |
| 159 | 3.1656810 | -1.1262746 | -3.6122599 |
| 160 | 2.0247636 | -1.9693127 | -1.6523434 |
| 161 | 1.2743616 | -0.5819890 | -1.4335409 |
| 162 | 2.0635839 | -1.7625071 | -0.9106692 |
| 163 | 0.5573015 | -0.5278260 | 0.3449202 |
| 164 | 1.3201986 | -1.3743276 | -3.3338290 |
| 165 | 2.0520504 | -1.9337126 | -1.1233537 |
| 166 | 0.1501339 | 0.9797831 | 2.3947355 |
| 167 | 1.8227365 | -1.4956895 | -0.1931979 |
| 168 | -0.4935315 | 1.7008095 | 1.5961480 |
| 169 | 1.9287011 | -1.4010257 | 0.1034533 |
| 170 | 0.4563414 | -0.4844630 | 0.3321659 |
| 171 | 1.0677781 | -1.2161786 | 0.1383304 |
| 172 | 1.4151285 | 0.1047705 | -0.1255630 |
| 173 | -0.0838008 | -0.2425371 | -0.1514705 |
| 174 | 0.3351227 | -1.4157810 | 0.0804488 |
| 175 | 1.4390622 | -1.0494598 | -0.3479998 |
| 176 | 1.3849537 | 0.2690570 | -0.3837815 |
| 177 | 0.7254943 | -0.4787162 | -0.8195401 |
| 178 | 1.6926268 | -0.2994484 | -1.9494934 |
| 179 | -0.9799874 | 1.0381485 | 0.3561739 |
| 180 | -1.0989264 | 1.0289251 | 0.0246007 |
| 181 | 0.9539940 | -1.9716752 | -0.3740614 |
| 182 | 0.8957322 | -1.2833299 | 3.0269591 |
| 183 | -0.8391831 | 0.0485813 | 0.0224010 |
| 184 | -2.3056544 | 1.2106311 | 1.6652628 |
| 185 | 0.9679084 | -0.4846706 | -0.7256451 |
| 186 | 1.2662664 | -0.7933096 | -0.3673744 |
| 187 | 1.0457403 | 0.1724144 | -1.1205701 |
| 188 | -1.5277792 | 1.1052131 | 2.3391888 |
| 189 | -0.1754647 | 0.7408755 | -0.5065606 |
| 190 | 1.2058601 | -0.2414466 | -0.5477483 |
| 191 | 0.6814702 | 0.6149574 | 1.0126469 |
| 192 | -0.8497423 | 1.3825465 | 0.1591452 |
| 193 | 0.1038677 | -0.5832026 | -0.6222342 |
| 194 | 0.7446903 | -0.1979301 | 1.3691859 |
| 195 | -0.3840437 | -0.6032295 | 0.0401704 |
| 196 | 1.7543011 | -1.0886484 | -0.6951447 |
| 197 | -0.1870959 | 0.1839153 | -0.4255816 |
| 198 | -0.8864783 | 1.3071363 | 1.3440563 |
| 199 | -0.0106591 | -0.1682833 | -0.8101959 |
| 200 | -0.2408353 | 0.3438407 | 1.4273740 |
| 201 | -0.0468628 | 0.3764746 | 0.1094338 |
| 202 | -0.8420080 | -0.1945396 | 0.4211585 |
| 203 | 0.0252447 | -2.1153360 | -1.7382737 |
| 204 | 0.5904463 | -0.9130674 | -0.7915262 |
| 205 | -0.5462648 | 0.3497093 | 1.2504308 |
| 206 | 0.0214590 | 0.4055771 | -1.0419466 |
| 207 | -0.6293104 | 0.5053261 | -0.0835213 |
| 208 | -0.8643903 | -1.7918671 | -0.8801539 |
| 209 | 0.2072623 | -0.6614889 | 1.2525696 |
| 210 | 1.1155970 | -0.3722728 | 0.3966079 |
| 211 | 0.3081642 | 0.1586610 | -0.8837296 |
| 212 | 1.3870540 | -0.6784124 | -0.6618193 |
| 213 | -1.4994738 | 0.0824890 | 0.5062311 |
| 214 | -0.7283932 | 1.2437401 | 0.4276507 |
| 215 | 0.6187606 | 0.8804955 | 1.1312596 |
| 216 | 1.7605258 | 0.5439587 | -1.9624287 |
| 217 | -0.3031339 | -0.7909051 | 0.2014529 |
| 218 | -0.5059877 | 0.4271447 | -0.3075433 |
| 219 | -1.8975496 | 0.9511143 | 0.3327584 |
| 220 | -0.6631145 | 0.8201171 | 0.2122263 |
| 221 | 0.4173146 | -1.8109460 | -1.5703982 |
| 222 | 0.9377251 | -0.2799136 | 0.1391862 |
| 223 | -0.8251228 | 0.6710683 | 0.0913491 |
| 224 | -2.0828422 | 1.5969725 | 2.1476909 |
| 225 | 0.0000442 | -0.7257318 | -0.3204829 |
| 226 | 0.2557414 | 0.9605752 | -0.2618041 |
| 227 | -1.7306036 | 0.9564396 | 1.3557451 |
| 228 | -1.3088122 | 1.2658477 | 0.3601722 |
| 229 | -0.6199446 | -0.2369562 | -0.9185620 |
| 230 | 0.7559797 | -1.5201141 | -0.6354062 |
| 231 | -1.3562039 | 0.3395534 | 1.6125185 |
| 232 | 1.1788169 | 1.3959739 | 0.0946064 |
| 233 | -0.2228757 | -1.1860622 | 0.6577650 |
| 234 | 0.9236243 | -0.4044715 | -0.2639062 |
| 235 | 0.7669923 | -0.2871061 | -0.7513423 |
| 236 | -0.9580882 | 0.9425857 | 0.6260820 |
| 237 | 2.3860484 | -1.0547386 | -0.5972318 |
| 238 | 1.8061842 | -0.6572775 | -0.5567167 |
| 239 | -0.4133801 | -0.4142889 | 1.0022967 |
| 240 | 0.1485323 | 0.4539667 | -0.5293854 |
| 241 | 1.3428984 | -0.5539540 | 0.2504764 |
| 242 | -1.2450330 | 0.3538685 | 0.3578461 |
| 243 | -0.5440004 | 0.1482858 | -1.4269887 |
| 244 | 0.6432294 | 0.3765119 | -0.1613056 |
| 245 | -1.7157603 | 1.5069871 | 1.0738804 |
| 246 | 1.7153394 | -0.9272950 | -0.8783255 |
| 247 | -0.3540130 | -0.5871412 | 0.0872690 |
| 248 | -1.9559826 | 2.4589257 | 1.0763947 |
| 249 | -2.1745707 | 0.6737419 | 1.3615485 |
| 250 | 1.6165741 | 0.3344244 | -0.9102370 |
| 251 | -1.9961950 | 1.6090419 | 2.0720183 |
| 252 | -0.1272891 | -1.1153993 | -1.8701366 |
| 253 | -0.9983739 | 0.0101301 | 0.8975034 |
| 254 | 0.5857948 | -1.0523579 | 1.5620424 |
| 255 | -0.8429583 | 0.4339232 | 0.7841885 |
| 256 | 1.5133899 | 0.6580896 | 0.9080449 |
| 257 | 1.8208861 | -1.4400733 | -1.3697890 |
| 258 | -0.6707887 | 0.4733612 | 0.3124379 |
| 259 | 0.0876472 | -0.8058521 | -0.6340125 |
| 260 | 2.1685144 | -0.7854659 | -0.7201728 |
| 261 | 0.9082819 | -1.1353506 | -1.6270345 |
| 262 | 0.7440821 | -0.9114252 | -0.7305909 |
| 263 | -1.7443183 | 0.0899663 | -0.3068435 |
| 264 | -0.7041372 | -0.2360947 | -0.8584979 |
| 265 | -2.0138454 | 1.6275791 | 0.3957977 |
| 266 | -2.3871472 | 0.5052006 | 1.3939842 |
| 267 | 0.0864558 | 0.0374373 | 0.9386799 |
| 268 | 0.3739279 | 0.3651652 | -0.2572919 |
| 269 | -0.3584666 | -1.3946008 | -0.5286457 |
| 270 | 1.2229180 | 0.6351347 | -0.3711065 |
| 271 | 0.9035632 | -0.4777527 | 0.2130855 |
| 272 | 0.3550738 | -0.8199743 | -1.1533472 |
| 273 | -0.6342262 | 1.2523842 | -0.1312231 |
| 274 | 0.8573957 | -1.3183499 | -0.2306841 |
| 275 | -0.9164994 | 0.5442612 | 0.6467203 |
| 276 | -0.5874836 | -0.1868970 | -0.6601317 |
| 277 | -1.2864169 | 0.6159636 | 1.4440184 |
| 278 | -1.3327523 | 0.5279876 | -0.3323753 |
| 279 | -0.3327827 | 0.2344553 | -1.0835801 |
| 280 | -0.6173638 | -0.1122981 | 0.9970415 |
| 281 | -0.0352932 | -1.3162969 | -0.9042622 |
| 282 | 0.6669514 | -0.3367675 | 0.4157708 |
| 283 | -0.4088405 | 0.4784102 | -0.4096794 |
| 284 | 0.2503699 | -1.0939116 | -1.0393495 |
| 285 | -0.2128611 | 1.2614056 | -0.3647840 |
| 286 | 0.1483441 | 0.0360152 | 0.3937545 |
| 287 | -3.0662083 | 1.8810672 | 1.9708598 |
| 288 | -1.0263377 | -0.8664550 | -0.5444923 |
| 289 | 1.4309696 | -0.9948478 | 0.9061844 |
| 290 | 1.8569767 | -1.5172336 | -0.4047097 |
| 291 | -0.4428215 | -0.0401771 | 1.0033186 |
| 292 | 1.9003993 | -1.7865378 | 0.1202371 |
| 293 | -2.3929112 | 0.9562462 | 1.5728841 |
| 294 | 1.4789944 | -0.2804958 | 1.0232953 |
| 295 | 0.0572745 | -0.4111873 | -0.8075252 |
| 296 | -0.2058447 | -0.4634304 | -0.3364237 |
| 297 | 0.9112563 | -2.4549370 | -1.1233474 |
| 298 | -1.1818031 | 0.3128863 | -0.1985622 |
| 299 | 0.4375730 | -0.7913382 | -1.0326057 |
| 300 | 0.2357914 | 0.2254650 | 1.2748681 |
| Parameter | Coefficient | SE | 95% CI | t(296) | p |
|---|---|---|---|---|---|
| (Intercept) | -9.64e-03 | 0.06 | (-0.12, 0.11) | -0.17 | 0.869 |
| aa anxiety | -0.58 | 0.06 | (-0.70, -0.46) | -9.61 | < .001 |
| aa avoidance | -0.37 | 0.06 | (-0.49, -0.25) | -6.09 | < .001 |
| aa anxiety × aa avoidance | -0.12 | 0.05 | (-0.21, -0.02) | -2.41 | 0.017 |
| aa_avoidance | Coefficient | SE | 95% CI | t(296) | p |
|---|---|---|---|---|---|
| -3.00 | -0.23 | 0.16 | (-0.55, 0.09) | -1.39 | 0.165 |
| -1.50 | -0.40 | 0.10 | (-0.60, -0.21) | -4.03 | < .001 |
| 0.00 | -0.58 | 0.06 | (-0.70, -0.46) | -9.61 | < .001 |
| 1.50 | -0.75 | 0.09 | (-0.93, -0.58) | -8.55 | < .001 |
| 3.00 | -0.93 | 0.15 | (-1.22, -0.63) | -6.20 | < .001 |
Marginal effects estimated for aa_anxiety
| aa_anxiety | Coefficient | SE | 95% CI | t(296) | p |
|---|---|---|---|---|---|
| -3.00 | -0.02 | 0.15 | (-0.32, 0.28) | -0.11 | 0.912 |
| -1.50 | -0.19 | 0.09 | (-0.37, -0.01) | -2.12 | 0.035 |
| 0.00 | -0.37 | 0.06 | (-0.49, -0.25) | -6.09 | < .001 |
| 1.50 | -0.54 | 0.10 | (-0.73, -0.35) | -5.52 | < .001 |
| 3.00 | -0.72 | 0.16 | (-1.03, -0.40) | -4.43 | < .001 |
Marginal effects estimated for aa_avoidance
Products are only one type of interaction (bilinear)
Measurement error gets compounded in moderation
Testing in the SEM framework may be necessary
Multivariate outliers can have a large influence
Power analysis is complicated for interactions
Interactions tend to be very power hungry
Simulation-based power analysis may be needed
Interpretations may differ between scales in GLM
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